A Construction of Polynomial Lattice Rules with Small Gain Coefficients

TL;DR

This paper develops polynomial lattice rules with small gain coefficients using a component-by-component method, achieving near-optimal variance decay rates for functions with bounded fractional variation, and compares their performance with digital nets.

Contribution

Introduces a component-by-component construction of polynomial lattice rules with small gain coefficients and analyzes their variance decay and computational efficiency.

Findings

Variance of estimators decays at rate $N^{-(2eta + 1) + abla}$ for functions with bounded fractional variation.

Constructed rules are nearly optimal for the specified function space.

Numerical results show improved performance of scrambled polynomial lattice rules over digital nets.

Abstract

In this paper we construct polynomial lattice rules which have, in some sense, small gain coefficients using a component-by-component approach. The gain coefficients, as introduced by Owen, indicate to what degree the method improves upon Monte Carlo. We show that the variance of an estimator based on a scrambled polynomial lattice rule constructed component-by-component decays at a rate of $N^{-(2\alpha + 1) +\delta}$, for all $\delta >0$, assuming that the function under consideration has bounded fractional variation of order $\alpha$ and where $N$ denotes the number of quadrature points. An analogous result is obtained for Korobov polynomial lattice rules. It is also established that these rules are almost optimal for the function space considered in this paper. Furthermore, we discuss the implementation of the component-by-component approach and show how to reduce the computational cost associated with it. Finally, we present numerical results comparing scrambled polynomial lattice rules and scrambled digital nets.